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52                        ANALYTICAL MECHANICS
Therefore the total frictional force and the total frictional torque are, respectively,
F = nal I *p de
Jo
and                        G=iM*lf'pde.
Jo
In order to carry out the integral of the foregoing expressions we have to make some assumption with regard to the nature
FIG. 38.
of dependence of p upon 6. But whatever the relation between p and 0 it is obvious that the sum, over all the surfaces of contact, of the vertical component of the normal reaction must equal the load which rests upon the bearings. If P denotes this load, then p must satisfy the condition
XA p sin 6  dA
= al IP sin 0 dO} JQ
where A is the total area of contact.
ILLUSTRATIVE EXAMPLE.
The normal pressure on the bearings is given by the relation p = p0 sin 6; find the total frictional force and the total frictional torque.